On the Distribution of Random Variables Corresponding to Musielak-Orlicz Norms

Journal article, 2013

Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X1,…,Xn are independent copies of X, then 1Cp∥x∥M≤𝔼∥(xiXi)ni=1∥p≤Cp∥x∥M, where Cp is a positive constant depending only on p. In case p=2 we need the function t↦tM′(t)−M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L1[0,1]. We also provide a general result replacing the ℓp-norm by an arbitrary N-norm. This complements some deep results obtained by Gordon, Litvak, Schütt, and Werner [Ann. Prob. 30 (2002)]. We also prove, in the spirit of that paper, a result which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak–Orlicz spaces.